Generalized robust toric ideals

TL;DR

This paper introduces a generalized concept of robustness for toric ideals, showing that for graph ideals, generalized robustness is characterized by equality of the universal Markov basis and the Graver basis, with implications for minimal generators.

Contribution

It extends the notion of robustness to generalized robustness for toric ideals and provides a graph-theoretic characterization for graph ideals.

Findings

A toric graph ideal is generalized robust iff its universal Markov basis equals its Graver basis.

Generalized robust graph ideals are characterized by properties of the graph's circuits.

Robust toric ideals have a unique minimal generating set, with all generators being indispensable.

Abstract

An ideal I is robust if its universal Gr\"obner basis is a minimal generating set for this ideal. In this paper, we generalize the meaning of robust ideals. An ideal is defined as generalized robust if its universal Gr\"obner basis is equal to its universal Markov basis. This article consists of two parts. In the first one, we study the generalized robustness on toric ideals of a graph G. We prove that a toric graph ideal is generalized robust if and only if its universal Markov basis is equal to the Graver basis of the ideal. Furthermore, we give a graph theoretical characterization of generalized robust graph ideals, which is based on terms of graph theoretical properties of the circuits of the graph G. In the second part, we go on to describe the general case of toric ideals, in which we prove that a robust toric ideal has a unique minimal system of generators, or in other words, all of its minimal generators are indispensable.